Towards an adaptive mesh refinement approach for stability and resolvent Analysis of external flows around complex geometries

We introduce an efficient and versatile numerical method to solve flow stability and resolvent analysis problems by combining the immersed boundary (IB) method, lattice Green's function (LGF), and adaptive mesh refinement (AMR). The immersed boundary method is used to represent the complex geometries without changing the underlying discretization of the PDE. The LGF ensures that we can accurately compute the stability and resolvent problems using the snuggest domain possible while maintaining the correct far-field boundary condition. Multilevel mesh is used to resolve different flow features on different length scales. Furthermore, we validated our algorithm by performing the 3D stability analysis of the flow past a rotating cylinder at various rotational rates. We also demonstrate the multilevel mesh by computing the stability problem of the flow past a cylinder with a control cylinder in its wake. Currently, we are working on leveraging the adaptive mesh refinement techniques to create tailored computational meshes to resolve the stability analysis problem for different flow configurations as accurately and as efficiently as possible. Preliminary results show that the AMR algorithm can track the vortical region of the unstable equilibrium even when it is not known a priori.